178 research outputs found
Perfect-information games with lower-semicontinuous payoffs
We prove that every multiplayer perfect-information game with bounded and lower-semicontinuous payoffs admits a subgame-perfect epsilon-equilibrium in pure strategies. This result complements Example 3 in Solan and Vieille [Solan, E., N. Vieille. 2003. Deterministic multi-player Dynkin games. J. Math. Econom. 39 911-929], which shows that a subgame-perfect epsilon-equilibrium in pure strategies need not exist when the payoffs are not lower-semicontinuous. In addition, if the range of payoffs is finite, we characterize in the form of a Folk Theorem the set of all plays and payoffs that are induced by subgame-perfect 0-equilibria in pure strategies
Infinite subgame perfect equilibrium in the Hausdorff difference hierarchy
Subgame perfect equilibria are specific Nash equilibria in perfect
information games in extensive form. They are important because they relate to
the rationality of the players. They always exist in infinite games with
continuous real-valued payoffs, but may fail to exist even in simple games with
slightly discontinuous payoffs. This article considers only games whose outcome
functions are measurable in the Hausdorff difference hierarchy of the open sets
(\textit{i.e.} when in the Baire space), and it characterizes the
families of linear preferences such that every game using these preferences has
a subgame perfect equilibrium: the preferences without infinite ascending
chains (of course), and such that for all players and and outcomes
we have . Moreover at
each node of the game, the equilibrium constructed for the proof is
Pareto-optimal among all the outcomes occurring in the subgame. Additional
results for non-linear preferences are presented.Comment: The alternative definition of the difference hierarchy has changed
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On values of repeated games with signals
We study the existence of different notions of value in two-person zero-sum
repeated games where the state evolves and players receive signals. We provide
some examples showing that the limsup value (and the uniform value) may not
exist in general. Then we show the existence of the value for any Borel payoff
function if the players observe a public signal including the actions played.
We also prove two other positive results without assumptions on the signaling
structure: the existence of the value in any game and the existence of
the uniform value in recursive games with nonnegative payoffs.Comment: Published at http://dx.doi.org/10.1214/14-AAP1095 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Doubly Reflected BSDEs and -Dynkin games: beyond the right-continuous case
We formulate a notion of doubly reflected BSDE in the case where the barriers
and do not satisfy any regularity assumption and with a general
filtration. Under a technical assumption (a Mokobodzki-type condition), we show
existence and uniqueness of the solution. In the case where is right
upper-semicontinuous and is right lower-semicontinuous, the solution is
characterized in terms of the value of a corresponding -Dynkin
game, i.e. a game problem over stopping times with (non-linear)
-expectation, where is the driver of the doubly reflected BSDE. In the
general case where the barriers do not satisfy any regularity assumptions, the
solution of the doubly reflected BSDE is related to the value of ''an
extension'' of the previous non-linear game problem over a larger set of
''stopping strategies'' than the set of stopping times. This characterization
is then used to establish a comparison result and \textit{a priori} estimates
with universal constants
Computer aided synthesis: a game theoretic approach
In this invited contribution, we propose a comprehensive introduction to game
theory applied in computer aided synthesis. In this context, we give some
classical results on two-player zero-sum games and then on multi-player non
zero-sum games. The simple case of one-player games is strongly related to
automata theory on infinite words. All along the article, we focus on general
approaches to solve the studied problems, and we provide several illustrative
examples as well as intuitions on the proofs.Comment: Invitation contribution for conference "Developments in Language
Theory" (DLT 2017
Regularity of the minmax value and equilibria in multiplayer Blackwell games
A real-valued function j that is defined over all Borel sets of a topological space is regular if for every Borel set W, j(W) is the supremum of j(C), over all closed sets C that are contained in W, and the infimum of j(O), over all open sets O that contain W. We study Blackwell games with finitely many players. We show that when each player has a countable set of actions and the objective of a certain player is represented by a Borel winning set, that player’s minmax value is regular. We then use the regularity of the minmax value to establish the existence of #-equilibria in two distinct classes of Blackwell games. One is the class of n-player Blackwell games where each player has a finite action space and an analytic winning set, and the sum of the minmax values over the players exceeds n 1. The other class is that of Blackwell games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values. For the latter class, we obtain a characterization of the set of equilibrium payoffs
On the Existence of Markov Perfect Equilibria in Perfect Information Games
We study the existence of pure strategy Markov perfect equilibria in two-person perfect information games. There is a state space X and each period player's possible actions are a subset of X. This set of feasible actions depends on the current state, which is determined by the choice of the other player in the previous period. We assume that X is a compact Hausdorff space and that
the action correspondence has an acyclic and asymmetric graph. For some states there may be no feasible actions and then the game ends. Payoffs are either discounted sums of utilities of the states visited, or the utility of the state where the game ends. We give sufficient conditions for the existence of equilibrium e.g. in case when either feasible action sets are finite or when players' payoffs are continuously dependent on each other. The latter class of games includes zero-sum games and pure coordination games.dynamic games, Markov perfect equilibrium
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